18.5 Heat Exchangers

The general function of a heat exchanger is to transfer heat from
one fluid to another. The basic component of a heat exchanger can be
viewed as a tube with one fluid running through it and another fluid
flowing by on the outside. There are thus three heat transfer
operations that need to be described:

Convective heat transfer
from fluid to the inner wall of the tube,

Conductive heat
transfer through the tube wall, and

Convective heat transfer from the
outer tube wall to the outside fluid.

Heat exchangers are typically classified according to flow
arrangement and type of construction. The simplest heat exchanger is
one for which the hot and cold fluids move in the same or opposite
directions in a concentric tube (or double-pipe) construction. In
the parallel-flow arrangement of
Figure 18.8(a), the hot and cold fluids
enter at the same end, flow in the same direction, and leave at the
same end. In the counterflow arrangement of
Figure 18.8(b), the fluids enter at
opposite ends, flow in opposite directions, and leave at opposite
ends.

Figure 18.8:
Concentric tubes heat exchangers

[Parallel flow]
[Counterflow]

Figure 18.9:
Cross-flow heat
exchangers.

[Finned with both
fluids unmixed.]
[Unfinned
with one fluid mixed and the other unmixed]

Alternatively, the fluids may be in cross flow (perpendicular to
each other), as shown by the finned and unfinned tubular heat
exchangers of Figure 18.9. The two
configurations differ according to whether the fluid moving over the
tubes is unmixed or mixed. In
Figure 18.9(a), the fluid is said to be
unmixed because the fins prevent motion in a direction (
) that is
transverse to the main flow direction (
). In this case the fluid
temperature varies with
and
. In contrast, for the unfinned
tube bundle of Figure 18.9(b), fluid
motion, hence mixing, in the transverse direction is possible, and
temperature variations are primarily in the main flow direction.
Since the tube flow is unmixed, both fluids are unmixed in the
finned exchanger, while one fluid is mixed and the other unmixed in
the unfinned exchanger.

To develop the methodology for heat exchanger analysis and design,
we look at the problem of heat transfer from a fluid inside a tube
to another fluid outside.

Figure 18.10:
Geometry for heat
transfer between two fluids

We examined this problem before in
Section 17.2 and found that the heat
transfer rate per unit length is given by

(18..21)

Here we have taken into account one additional thermal resistance
than in Section 17.2, the resistance
due to convection on the interior, and include in our expression for
heat transfer the bulk temperature of the fluid,
, rather than
the interior wall temperature,
.

It is useful to define an overall heat transfer coefficient
per unit length as

The objective is to find the mean temperature of the fluid at
,
, in the case where fluid comes in at
with temperature
and leaves at
with temperature
. The expected
distribution for heating and cooling are sketched in
Figure 18.12.

For heating (
), the heat flow from the pipe wall in a
length
is

where
is the pipe diameter. The heat given to the fluid (the
change in enthalpy) is given by

where
is the density of the fluid,
is the mean velocity
of the fluid,
is the specific heat of the fluid and
is the mass flow rate of the fluid. Setting the last two expressions
equal and integrating from the start of the pipe, we find

The concept of a logarithmic mean temperature difference is useful
in the analysis of heat exchangers. We will define a logarithmic
mean temperature difference for the general counterflow heat
exchanger below.

We return to our original problem, to
Figure 18.11, and write an
overall heat balance between the two counterflowing streams as

From a local heat balance, the heat given up by stream
in length
x is
. (There is a negative sign since
decreases). The heat taken up by stream
is
. (There is a negative sign because
decreases as
increases). The local heat balance is